Instructions
This worksheet challenges you to solve word problems by converting the story into mathematical equations. Follow these steps for each section:
- Define the Variable: Decide what unknown quantity the question is asking for, and assign it a letter (like $x$ or $y$).
- Write the Equation: Translate the words into a balanced equation (an expression with an equals sign).
- Solve: Use inverse operations to find the value of the variable.
- Check: Ensure your answer makes sense in the context of the original story.
Section 1: Translating Words into Expressions
Write the algebraic expression that matches the phrase. (Example: 'Four more than a number' $\rightarrow x + 4$)
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Nine times a number, decreased by two. Expression: _____
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The cost of a pizza, $p$, split equally among four friends. Expression: _____
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Thirty less than six times a number $n$. Expression: _____
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A number $y$ added to fifteen, and then the total multiplied by three. Expression: _____
Section 2: Identify and Set Up (No Solving Needed Yet)
For the problems below, identify the variable and write the full equation, but do not solve it.
| Word Problem | Variable Defined (e.g., Let $x$ = cost) | Algebraic Equation |
|---|---|---|
| Example: Maya bought 3 shirts and spent $45 total. Write an equation to find the cost, $c$, of one shirt. | Let $c$ = cost of one shirt. | $3c = 45$ |
| 1. Liam spent $12 on movie tickets and $8 on popcorn. He started with $30. How much money, $m$, did he have left? | ||
| 2. A rectangle has a perimeter of 40 inches. The length is 12 inches. Write an equation to find the width, $w$. (Perimeter = $2L + 2W$) | ||
| 3. Ms. Perez gave 4 pencils to each of her $c$ students. If she gave out 108 pencils total, how many students are in the class? | ||
| 4. A taxi charges a flat fee of $5 plus $2 for every mile, $m$. If the total fare was $19, how many miles were driven? | ||
| 5. If you increase a number $y$ by 5 and then multiply the result by 2, the total is 34. Find the number. |
Section 3: Solve the Story Problems
Read each scenario carefully. Define your variable, set up the equation, solve, and write your final answer in a complete sentence.
Problem A: The Arcade Challenge
At the Mega-Arcade, tokens cost $0.50 each. You spend $15.50 total. This total included a $2.50 entrance fee. How many tokens, $t$, did you buy?
- Define Variable: Let $t$ = _____
- Write Equation: _____
- Solve: $t$ = _____
- Final Answer: _____
Problem B: The Running Track
Eliza ran three times as far as her friend Noah. If Eliza ran 9.6 miles, how far, $d$, did Noah run?
- Define Variable: Let $d$ = _____
- Write Equation: _____
- Solve: $d$ = _____
- Final Answer: _____
Problem C: The Geometry Riddle
The area of a rectangular garden is 96 square feet. If the width of the garden is 8 feet, what is the length, $L$? (Area = Length $\times$ Width)
- Define Variable: Let $L$ = _____
- Write Equation: _____
- Solve: $L$ = _____
- Final Answer: _____
Section 4: The Advanced Challenge
This problem requires combining terms and setting up a multi-step equation.
Challenge Problem: The Savings Goal
Marco is saving money for a new video game that costs $65. He already has $17 saved. He decides to save $8 every week, $w$, until he reaches his goal.
A. Write an equation that represents the total cost, $65, based on the amount saved, $17, and the weekly savings, $8w$.
Equation: _____
B. How many weeks will it take Marco to save enough money?
Show your work:
C. If Marco waited 6 weeks, would he have enough money? Explain why or why not using mathematics.
Answer Key
Section 1: Translating Words into Expressions
- $9x - 2$ (or $9n - 2$, if $n$ is used)
- $p / 4$
- $6n - 30$
- $3(y + 15)$
Section 2: Identify and Set Up
| Word Problem | Variable Defined | Algebraic Equation |
|---|---|---|
| 1. Liam spent $12 on movie tickets and $8 on popcorn. He started with $30. How much money, $m$, did he have left? | Let $m$ = money left | $30 - 12 - 8 = m$ (or $30 - (12 + 8) = m$) |
| 2. A rectangle has a perimeter of 40 inches. The length is 12 inches. Write an equation to find the width, $w$. | Let $w$ = width in inches | $2(12) + 2w = 40$ (or $24 + 2w = 40$) |
| 3. Ms. Perez gave 4 pencils to each of her $c$ students. If she gave out 108 pencils total, how many students are in the class? | Let $c$ = number of students | $4c = 108$ |
| 4. A taxi charges a flat fee of $5 plus $2 for every mile, $m$. If the total fare was $19, how many miles were driven? | Let $m$ = miles driven | $5 + 2m = 19$ |
| 5. If you increase a number $y$ by 5 and then multiply the result by 2, the total is 34. Find the number. | Let $y$ = the number | $2(y + 5) = 34$ |
Section 3: Solve the Story Problems
Problem A: The Arcade Challenge
- Define Variable: Let $t$ = number of tokens bought.
- Write Equation: $2.50 + 0.50t = 15.50$
- Solve: $0.50t = 13.00$; $t = 26$
- Final Answer: You bought 26 tokens.
Problem B: The Running Track
- Define Variable: Let $d$ = distance Noah ran (in miles).
- Write Equation: $3d = 9.6$
- Solve: $d = 9.6 / 3$; $d = 3.2$
- Final Answer: Noah ran 3.2 miles.
Problem C: The Geometry Riddle
- Define Variable: Let $L$ = length of the garden (in feet).
- Write Equation: $L \times 8 = 96$ (or $8L = 96$)
- Solve: $L = 96 / 8$; $L = 12$
- Final Answer: The length of the garden is 12 feet.
Section 4: The Advanced Challenge
Challenge Problem: The Savings Goal
A. Equation: $17 + 8w = 65$
B. Show your work: $17 + 8w = 65$ $8w = 65 - 17$ $8w = 48$ $w = 6$ It will take Marco 6 weeks.
C. If Marco waited 6 weeks, would he have enough money? Explain why or why not using mathematics.
Yes, he would have exactly enough money. Since the calculation in Part B showed $w=6$ is the solution, after 6 weeks he will have saved $17 + 8(6) = 17 + 48 = $65.